Done for 1st iteration

Manuscript/BSE_JPCL.bib

@@ -1,13 +1,24 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-06-05 12:56:19 +0200 
+%% Created for Pierre-Francois Loos at 2020-06-05 21:33:09 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Wu_2019,
+	Author = {Xin‐Ping Wu and Indrani Choudhuri  and Donald G. Truhlar},
+	Date-Added = {2020-06-05 20:35:01 +0200},
+	Date-Modified = {2020-06-05 20:41:08 +0200},
+	Doi = {10.1002/eem2.12051},
+	Journal = {Energy Environ. Mat.},
+	Pages = {251--263},
+	Title = {Computational Studies of Photocatalysis with Metal--Organic Frameworks},
+	Volume = {2},
+	Year = {2019}}
+
 @phdthesis{Rebolini_PhD,
 	Author = {E. Rebolini},
 	Date-Added = {2020-05-25 11:57:30 +0200},

Manuscript/BSE_JPCL.tex

@@ -15,6 +15,7 @@
 \usepackage{newtxtext,newtxmath}
 
 \usepackage{pifont}
+\usepackage{float}
 \usepackage{graphicx}
 \usepackage{dcolumn}
 \usepackage{multirow}
@@ -228,7 +229,7 @@ In its press release announcing the attribution of the 2013 Nobel prize in Chemi
 Simulations are so realistic that they predict the outcome of traditional experiments.''} \cite{Nobel_2003} 
 Martin Karplus' Nobel lecture moderated this statement, introducing his presentation by a 1929 quote from Dirac emphasizing that laws of quantum mechanics are \textit{``much too complicated to be soluble''}, urging scientists to develop \textit{``approximate practical methods''}. This is where the electronic structure community stands, attempting to develop robust approximations to study with increasing accuracy the properties of ever more complex systems. 
 
-The study of neutral electronic excitations in condensed-matter systems, from molecules to extended solids, has witnessed the development of a large number of such approximate methods with numerous applications to a large variety of fields, from the  prediction of the colour of precious metals  for jewellery, \cite{Prandini_2019} to the understanding, \eg, of the basic principles behind organic photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. \cite{Kippelen_2009,Improta_2016}  \xavier{[Xav: Good ref on theory for photocatalysis still needed.]}
+The study of neutral electronic excitations in condensed-matter systems, from molecules to extended solids, has witnessed the development of a large number of such approximate methods with numerous applications to a large variety of fields, from the  prediction of the colour of precious metals  for jewellery, \cite{Prandini_2019} to the understanding, \eg, of the basic principles behind organic photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. \cite{Kippelen_2009,Improta_2016,Wu_2019}
 The present \textit{Perspective} aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} that, while sharing many features with time-dependent density-functional theory (TD-DFT), \cite{Runge_1984} including computational scaling with system size, relies on a very different formalism, with specific difficulties but also potential solutions to known issues. \cite{Blase_2018}
 \\
 
@@ -303,7 +304,7 @@ that can be regarded as the lowest-order perturbation in terms of the screened C
 where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulomb potential. 
 
 %%% FIG 1 %%%
-\begin{figure}
+\begin{figure}[h]
 	\includegraphics[width=0.55\linewidth]{fig1/fig1}
 	\caption{
 	Hedin's pentagon connects the Green's function $G$, its non-interacting analog $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb potential $W$, and the self-energy $\Sigma$ through a set of five integro-differential equations known as Hedin's equations. \cite{Hedin_1965}
@@ -479,7 +480,7 @@ with the experimental (photoemission) fundamental gap
 where $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$ are the ionization potential and the electron affinity of the $\Nel$-electron system (see Fig.~\ref{fig:gaps}).
 
 %%% FIG 2 %%%
-\begin{figure*}
+\begin{figure*}[h]
 	\includegraphics[width=0.7\linewidth]{gaps}
 	\caption{
 	Definition of the optical gap $\EgOpt$ and fundamental gap $\EgFun$.
@@ -511,15 +512,6 @@ As a result, BSE singlet excitation energies starting from such improved quasipa
 For sake of illustration, an average error of $0.2$ eV was found for the well-known Thiel set \cite{Schreiber_2008} gathering more than hundred representative singlet excitations from a large variety of representative molecules. \cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017} 
 This is equivalent to the best TD-DFT results obtained by scanning a large variety of global hybrid functionals with various amounts of exact exchange. 
 
-%%% FIG 3 %%%
-\begin{figure}
-	\includegraphics[width=6cm]{CTfig}
-	\caption{
-		Symbolic representation of (Top) extended Wannier exciton with large electron-hole average distance, and (Bottom) Frenkel and charge-transfer (CT) excitations at a donor-acceptor interface. Wannier and CT excitations require long-range electron-hole interaction accounting for the dielectric constant of the host. In the case of Wannier exciton, the binding energy $E_B$ can be well approximated by the standard hydrogenoid model with $\mu$ the effective mass and $\epsilon$ the dielectric constant.
-	\label{fig:CTfig}}
-\end{figure} 
-%%% %%% %%%
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Successes \& Challenges} 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -528,34 +520,46 @@ This is equivalent to the best TD-DFT results obtained by scanning a large varie
 %==========================================
 \subsection{Charge-transfer excited states} 
 %==========================================
-A very remarkable success of the BSE formalism lies in the description of charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT adopting standard (semi-)local or hybrid functionals. \cite{Dreuw_2004} 
+A very remarkable success of the BSE formalism lies in the description of charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT adopting standard (semi-)local functionals. \cite{Dreuw_2004} 
 Similar difficulties emerge in solid-state physics for semiconductors where extended Wannier excitons, characterized by weakly overlapping electrons and holes (Fig.~\ref{fig:CTfig}), cause a dramatic deficit of spectral weight at low energy. \cite{Botti_2004} 
 These difficulties can be ascribed to the lack of long-range electron-hole interaction with local xc functionals.
 It can be cured through an exact exchange contribution, a solution that explains in particular the success of optimally-tuned range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012} 
 The analysis of the screened Coulomb potential matrix elements in the BSE kernel [see Eq.~\eqref{eq:BSEkernel}] reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc) where screening reduces the long-range electron-hole interactions. 
 The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Rocca_2010,Cudazzo_2010,Lastra_2011,Blase_2011,Baumeier_2012a,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2013,Yin_2014} opening the way to important applications such as doping, \cite{Li_2017b}  photovoltaics or photocatalysis in organic systems.\\
 
+%%% FIG 3 %%%
+\begin{figure}[h]
+	\includegraphics[width=0.6\linewidth]{CTfig}
+	\caption{
+		Symbolic representation of extended Wannier exciton with large electron-hole average distance (top), and Frenkel and charge-transfer (CT) excitations at a donor-acceptor interface (bottom). 
+		Wannier and CT excitations require long-range electron-hole interaction accounting for the host dielectric constant. 
+		In the case of Wannier excitons, the binding energy $\EB$ can be well approximated by the standard hydrogenoid model where $\mu$ is the effective mass and $\epsilon$ is the dielectric constant.
+	\label{fig:CTfig}}
+\end{figure} 
+%%% %%% %%%
+
 %==========================================
 \subsection{Combining BSE with PCM and QM/MM models} 
 %==========================================
 
-The ability to account for the effect on the excitation energies of an electrostatic and dielectric environment (an electrode, a solvent, a molecular interface, etc.) is an important step towards the description of realistic systems. Pioneering  $BSE$ studies demonstrated e.g. the large renormalization of charge and neutral excitations of molecular systems and nanotubes close to a metallic electrode or in bundles. \cite{Lastra_2011,Rohlfing_2012,Spataru_2013}
+The ability to account for the effect on the excitation energies of an electrostatic and dielectric environment (an electrode, a solvent, a molecular interface, \ldots) is an important step towards the description of realistic systems. 
+Pioneering BSE studies demonstrated, for example, the large renormalization of charged and neutral excitations in molecular systems and nanotubes close to a metallic electrode or in bundles. \cite{Lastra_2011,Rohlfing_2012,Spataru_2013}
 Recent attempts to merge the $GW$ and BSE formalisms with model polarizable environments at the PCM or QM/MM levels
-\cite{Baumeier_2014,Duchemin_2016,Li_2016,Varsano_2016,Duchemin_2018,Li_2019,Tirimbo_2020} paved the way not only to interesting applications but also to a better understanding of the merits of these approaches relying on the use of the  screened Coulomb potential designed to capture  polarization effects at all spatial ranges. As a matter of fact,
+\cite{Baumeier_2014,Duchemin_2016,Li_2016,Varsano_2016,Duchemin_2018,Li_2019,Tirimbo_2020} paved the way not only to interesting applications but also to a better understanding of the merits of these approaches relying on the use of the screened Coulomb potential designed to capture polarization effects at all spatial ranges. 
-dressing the bare Coulomb potential with the reaction field matrix 
+As a matter of fact, dressing the bare Coulomb potential with the reaction field matrix 
-$ \;[
+$[
-	v(\br,\br') \longrightarrow v(\br,\br') + v^{\text{reac}}(\br,\br'; \omega)
+v(\br,\br') \longrightarrow v(\br,\br') + v^{\text{reac}}(\br,\br'; \omega)
 ]$
- in the relation between  the screened Coulomb potential $W$ and the independent-electron susceptibility [see Eq.~\eqref{eq:defW}] allows to perform $GW$ and BSE calculations in a polarizable environment (a solvent, a donor/acceptor interface, a semiconducting or metallic substrate, etc) with the same complexity as in the gas phase. 
+in the relation between  the screened Coulomb potential $W$ and the independent-electron susceptibility [see Eq.~\eqref{eq:defW}] allows to perform $GW$ and BSE calculations in a polarizable environment (a solvent, a donor/acceptor interface, a semiconducting or metallic substrate, etc) with the same complexity as in the gas phase. 
 The reaction field matrix $v^{\text{reac}}(\br,\br'; \omega)$ describes the potential generated in $\br'$ by the charge rearrangements in the polarizable environment induced by a source charge located in $\br$, where $\br$ and $\br'$ lie in the quantum mechanical subsystem of interest. 
 The reaction field is dynamical since the dielectric properties of the environment, such as the macroscopic dielectric constant $\epsilon_M(\omega)$, are in principle frequency dependent.   
-Once the reaction field matrix is known, with typically $\order*{\Norb N_{MM}^2}$ operations (where $\Norb$ is the number of  orbitals and $N_{MM}$ the number of polarizable atoms in the environment), the full spectrum of $GW$ quasiparticle energies and BSE neutral excitations can be renormalized by the effect of the environment.
+Once the reaction field matrix is known, with typically $\order*{\Norb N_\text{MM}^2}$ operations (where $\Norb$ is the number of  orbitals and $N_\text{MM}$ the number of polarizable atoms in the environment), the full spectrum of $GW$ quasiparticle energies and BSE neutral excitations can be renormalized by the effect of the environment.
 
 A remarkable property \cite{Duchemin_2018} of the scheme described above, which combines the BSE formalism with a polarizable environment, is that the renormalization of the electron-electron and electron-hole interactions by the reaction field   captures both linear-response and state-specific contributions \cite{Cammi_2005} to the solvatochromic shift of the optical lines, allowing to treat on the same footing Frenkel and CT excitations. 
 This is an important advantage as compared to, \eg, TD-DFT where linear-response and state-specific effects have to be explored with different formalisms. 
 
-To date, environmental effects on fast electronic excitations are only included by considering the low-frequency optical response of the polarizable medium (\eg,  considering the $\epsilon_{\infty} \simeq 1.78$ macroscopic dielectric constant for water in the optical range), neglecting the frequency dependence of the dielectric constant in the optical range. 
+To date, environmental effects on fast electronic excitations are only included by considering the low-frequency optical response of the polarizable medium (\eg,  considering the $\epsilon_{\infty} \simeq 1.78$ macroscopic dielectric constant of water in the optical range), neglecting the frequency dependence of the dielectric constant in the optical range. 
-Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, \ie, in situations where neither the adiabatic nor antiadiabatic limits are expected to be valid (for a recent discussion, see Ref.
+Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, \ie, in situations where neither the adiabatic limit nor the antiadiabatic limit are expected to be valid (for a recent discussion, see Ref.
 ~\citenum{Huu_2020}).
 
 We now leave the description of successes to discuss difficulties and future directions of developments and improvements.
@@ -566,18 +570,18 @@ We now leave the description of successes to discuss difficulties and future dir
 %==========================================
 As emphasized above, the BSE eigenvalue equation in the single-excitation space [see Eq.~\eqref{eq:BSE-eigen}] is formally equivalent to that of TD-DFT or TD-HF. \cite{Dreuw_2005}
 Searching iteratively for the lowest eigenstates exhibits the same $\order*{\Norb^4}$ matrix-vector multiplication computational cost within BSE and TD-DFT. 
-Concerning the construction of the BSE Hamiltonian, it is no more expensive than building its TD-DFT analogue with hybrid functionals, reducing again to $\order*{\Norb^4}$ operations with standard RI techniques. Explicit calculation of the full BSE Hamiltonian in  transition space can be further avoided using density matrix perturbation theory,
+Concerning the construction of the BSE Hamiltonian, it is no more expensive than building its TD-DFT analogue with hybrid functionals, reducing again to $\order*{\Norb^4}$ operations with standard RI techniques. 
-\cite{Rocca_2010,Nguyen_2019} not reducing though the $\order*{\Norb^4}$ scaling, sacrificing further the knowledge of the eigenvectors. 
+Explicit calculation of the full BSE Hamiltonian in transition space can be further avoided using density matrix perturbation theory, \cite{Rocca_2010,Nguyen_2019} not reducing though the $\order*{\Norb^4}$ scaling, but sacrificing further the knowledge of the eigenvectors. 
-Exploiting further the locality of localized atomic basis orbitals, the BSE absorption spectrum could be obtained with $\order*{\Norb^3}$ operations using such iterative techniques. \cite{Ljungberg_2015} 
+Exploiting further the locality of the atomic orbital basis, the BSE absorption spectrum can be obtained with $\order*{\Norb^3}$ operations using such iterative techniques. \cite{Ljungberg_2015} 
 With the same restriction on the eigenvectors, a time-propagation approach, similar to that implemented for TD-DFT, \cite{Yabana_1996} combined with stochastic techniques to reduce the cost of building the BSE Hamiltonian matrix elements, allows quadratic scaling with systems size. \cite{Rabani_2015}
 
 In practice, the main bottleneck for standard BSE calculations as compared to TD-DFT resides in the preceding $GW$ calculation that scales as $\order{\Norb^4}$ with system size using plane-wave basis sets or RI techniques, but with a rather large prefactor. 
 %%Such a cost is mainly associated with calculating the free-electron susceptibility with its entangled summations over occupied and virtual states. 
 %%While attempts to bypass the $GW$ calculations are emerging, replacing quasiparticle energies by Kohn-Sham eigenvalues matching energy electron addition/removal, \cite{Elliott_2019} 
 The field of low-scaling $GW$ calculations is however witnessing significant advances. 
- While the sparsity of e.g. the overlap matrix in the atomic basis allows to reduce the scaling in the large size limit,  \cite{Foerster_2011,Wilhelm_2018} efficient real-space-grid and time techniques are blooming, \cite{Rojas_1995,Liu_2016} borrowing in particular the well-known Laplace transform approach used in quantum chemistry. \cite{Haser_1992}
+While the sparsity of, for example, the overlap matrix in the atomic orbital basis allows to reduce the scaling in the large size limit, \cite{Foerster_2011,Wilhelm_2018} efficient real-space grids and time techniques are blooming, \cite{Rojas_1995,Liu_2016} borrowing in particular the well-known Laplace transform approach used in quantum chemistry. \cite{Haser_1992}
 Together with a stochastic sampling of virtual states, this family of techniques allow to set up linear scaling $GW$ calculations. \cite{Vlcek_2017} 
-The separability of occupied and virtual states summations lying at the heart of these approaches are now spreading fast in quantum chemistry within the interpolative separable density fitting (ISDF) approach applied to calculating with cubic scaling the susceptibility needed in random-phase approximation (RPA) and $GW$ calculations. \cite{Lu_2017,Duchemin_2019,Gao_2020}   
+The separability of occupied and virtual states summations lying at the heart of these approaches are now spreading fast in quantum chemistry within the interpolative separable density fitting (ISDF) approach applied for calculating with cubic scaling the susceptibility needed in random-phase approximation (RPA) and $GW$ calculations. \cite{Lu_2017,Duchemin_2019,Gao_2020}   
 These ongoing developments pave the way to applying the $GW$@BSE formalism to systems containing several hundred atoms on standard laboratory clusters.
 \\
 
@@ -585,11 +589,11 @@ These ongoing developments pave the way to applying the $GW$@BSE formalism to sy
 \subsection{The triplet instability challenge}
 %==========================================
 The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or 
-stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels, \titou{[REFS]}
+stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels, contaminating as well TD-DFT calculations with popular range-separated hybrids that generally contains a large fraction of exact exchange in the long-range.  
- contaminating as well TD-DFT calculations with popular range-separated hybrids that generally contains a large fraction of exact exchange in the long-range.  
 While TD-DFT with range-separated hybrids can benefit from tuning the range-separation parameter(s) as a mean to act on the triplet instability, \cite{Sears_2011} BSE calculations do not offer this pragmatic way-out since the screened Coulomb potential that builds the kernel does not offer any parameter to tune.
 
-Benchmark calculations  \cite{Jacquemin_2017b,Rangel_2017} clearly concluded that triplets are notably too low in energy within BSE and that the use of the Tamm-Dancoff approximation was able to partly reduce this error. However, as it stands, the BSE inaccuracy for triplets remains rather unsatisfactory for reliable applications. 
+Benchmark calculations \cite{Jacquemin_2017b,Rangel_2017} clearly concluded that triplets are notably too low in energy within BSE and that the use of the Tamm-Dancoff approximation was able to partly reduce this error. 
+However, as it stands, the BSE accuracy for triplets remains rather unsatisfactory for reliable applications. 
 An alternative cure was offered by hybridizing TD-DFT and BSE, that is, by adding to the BSE kernel the correlation part of the underlying DFT functional used to build the susceptibility and resulting screened Coulomb potential $W$. \cite{Holzer_2018b}
 \\
 
@@ -600,19 +604,19 @@ An alternative cure was offered by hybridizing TD-DFT and BSE, that is, by addin
 %
 %This points to another direction for the BSE formalism, namely the calculation of GS total energy with the correlation energy calculated at the BSE level. Such a task was performed by several groups using in particular the adiabatic connection fluctuation-dissipation theorem (ACFDT), focusing in particular on small dimers. \cite{Olsen_2014,Holzer_2018b,Li_2020,Loos_2020}\\
 
-The features of ground- and excited-state potential energy surfaces (PES) are critical for the faithful description and a deeper understanding of photochemical and photophysical processes. \cite{Bernardi_1996,Olivucci_2010,Robb_2007}
+The features of ground- and excited-state potential energy surfaces (PES) are critical for the faithful description and a deeper understanding of photochemical and photophysical processes. \cite{Olivucci_2010}
 For example, chemoluminescence, fluorescence and other related processes are associated with geometric relaxation of excited states, and structural changes upon electronic excitation. \cite{Navizet_2011}
 Reliable predictions of these mechanisms, which have attracted much experimental and theoretical interest lately, require exploring the ground- and excited-state PES. 
-From a theoretical point of view, the accurate prediction of excited electronic states remains a challenge, \cite{Gonzales_2012, Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Andersson_1990,Andersson_1992,Roos_1996,Angeli_2001}) cannot be afforded.
+From a theoretical point of view, the accurate prediction of excited electronic states remains a challenge, \cite{Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Roos_1996,Angeli_2001}) cannot be afforded.
 For the last two decades, TD-DFT has been the go-to method to compute absorption and emission spectra in large molecular systems.
 
 In TD-DFT, the PES for the excited states can be easily and efficiently obtained as a function of the molecular geometry by simply adding the ground-state DFT energy to the excitation energy of the selected state. 
-One of the strongest assets of TD-DFT is the availability of first- and second-order analytic nuclear gradients (\ie, the first derivatives of the excited-state energy with respect to the nuclear displacements), which enables the exploration of excited-state PES.\cite{Furche_2002}
+One of the strongest assets of TD-DFT is the availability of first- and second-order analytic nuclear gradients (\ie, the first- and second-order derivatives of the excited-state energy with respect to the nuclear displacements), which enables the exploration of excited-state PES. \cite{Furche_2002}
 
 A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytical nuclear gradients for both the ground and excited states, preventing efficient studies of excited-state processes.
 While calculations of the $GW$ quasiparticle energy ionic gradients is becoming increasingly popular,
 \cite{Lazzeri_2008,Faber_2011b,Yin_2013,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003} 
-In this seminal work devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, with the approximation that the gradient of the screened Coulomb potential can be neglected,  computing further the KS-LDA forces as its ground-state contribution.
+In this seminal work devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, with the approximation that the gradient of the screened Coulomb potential can be neglected, computing further the KS-LDA forces as its ground-state contribution.
 \\
 
 %==========================================
@@ -622,13 +626,13 @@ In this seminal work devoted to small molecules (\ce{CO} and \ce{NH3}), only the
 In contrast to TD-DFT which relies on KS-DFT as its ground-state analog, the ground-state BSE energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
 Consequently, the BSE ground-state formalism remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020,Loos_2020}
 
-A promising route, which closely follows RPA-type formalisms, \cite{Furche_2008,Toulouse_2009,Toulouse_2010,Angyan_2011,Ren_2012a} is to calculated the ground-state BSE energy within the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework. \cite{Furche_2005,Olsen_2014,Maggio_2016,Holzer_2018,Loos_2020}
+A promising route, which closely follows RPA-type formalisms, \cite{Angyan_2011} is to calculated the ground-state BSE energy within the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework. \cite{Furche_2005}
 Thanks to comparisons with both similar and state-of-art computational approaches, it was recently shown that the ACFDT@BSE@$GW$ approach yields extremely accurate PES around equilibrium, and can even compete with high-order coupled cluster methods in terms of absolute ground-state energies and equilibrium distances. \cite{Loos_2020}
 Their accuracy near the dissociation limit remains an open question. \cite{Caruso_2013,Olsen_2014,Colonna_2014,Hellgren_2015,Holzer_2018}
 Indeed, in the largest available benchmark study \cite{Holzer_2018} encompassing the total energies of the atoms \ce{H}--\ce{Ne}, the atomization energies of the 26 small molecules forming the HEAT test set, and the bond lengths and harmonic vibrational frequencies of $3d$ transition-metal monoxides, the BSE correlation energy, as evaluated within the ACFDT framework, \cite{Furche_2005} was mostly discarded from the set of tested techniques due to instabilities (negative frequency modes in the BSE polarization propagator) and replaced by an approximate (RPAsX) approach where the screened-Coulomb potential matrix elements was removed from the resonant electron-hole contribution. \cite{Maggio_2016,Holzer_2018} 
 Moreover, it was also observed in Ref.~\citenum{Loos_2020} that, in some cases, unphysical irregularities on the ground-state PES show up due to the appearance of discontinuities as a function of the bond length for some of the $GW$ quasiparticle energies. 
-Such an unphysical behavior stems from defining the quasiparticle energy as the solution of the quasiparticle equation with the largest spectral weight in cases where several solutions can be found.
+Such an unphysical behavior stems from defining the quasiparticle energy as the solution of the quasiparticle equation with the largest spectral weight in cases where several solutions can be found [see Eq.~\eqref{eq:QP-eq}].
-This shortcoming has been thoroughly described in several previous studies.\cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
+We refer the interested reader to Refs.~\citenum{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020} for a detailed discussion of this particular point.
 \\
 
 %==========================================
@@ -656,8 +660,8 @@ This shortcoming has been thoroughly described in several previous studies.\cite
 %==========================================
 \subsection{Beyond the static approximation}
 %==========================================
-Going beyond the static approximation is a difficult challenge which has been, nonetheless, embraced by several groups.\cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
+Going beyond the static approximation is a difficult challenge which has been, nonetheless, embraced by several groups.\cite{Strinati_1988,Rohlfing_2000,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019}
-As mentioned earlier in this \textit{Perspective}, most of BSE calculations are performed within the so-called static approximation, which substitutes the dynamically-screened (\ie, frequency-dependent) Coulomb potential $W(\omega)$ by its static limit $W(\omega = 0)$.
+As mentioned earlier in this \textit{Perspective}, most of BSE calculations are performed within the so-called static approximation, which substitutes the dynamically-screened (\ie, frequency-dependent) Coulomb potential $W(\omega)$ by its static limit $W(\omega = 0)$ [see Eq.~\eqref{eq:Wmatel}].
 It is important to mention that diagonalizing the BSE Hamiltonian in the static approximation corresponds to solving a \textit{linear} eigenvalue problem in the space of single excitations, while it is, in its dynamical form, a non-linear eigenvalue problem (in the same space) which is much harder to solve from a numerical point of view.
 In complete analogy with the ubiquitous adiabatic approximation in TD-DFT, one key consequence of the static approximation is that double (and higher) excitations are completely absent from the BSE spectrum, which obviously hampers the applicability of BSE as double excitation may play, indirectly, a key role in photochemistry mechanisms. 
 Higher excitations would be explicitly present in the BSE Hamiltonian by ``unfolding'' the dynamical BSE kernel, and one would recover a linear eigenvalue problem with, nonetheless, a much larger dimension.
@@ -669,7 +673,7 @@ Thanks to first-order perturbation theory, Rohlfing and coworkers have developed
 With such a scheme, they have been able to compute the excited states of biological chromophores, showing that taking into account the electron-hole dynamical screening is important for an accurate description of the lowest $n \ra \pi^*$ excitations. \cite{Ma_2009a,Ma_2009b,Baumeier_2012b}
 Studying PYP, retinal and GFP chromophore models, Ma \textit{et al.}~found that \textit{``the influence of dynamical screening on the excitation energies is about $0.1$ eV for the lowest $\pi \ra \pi^*$ transitions, but for the lowest $n \ra \pi^*$ transitions the influence is larger, up to $0.25$ eV.''} \cite{Ma_2009b}
 Zhang \textit{et al.}~have studied the frequency-dependent second-order Bethe-Salpeter kernel and they have observed an appreciable improvement over configuration interaction with singles (CIS), time-dependent Hartree-Fock (TDHF), and adiabatic TD-DFT results. \cite{Zhang_2013}
-Rebolini and Toulouse have performed a similar investigation in a range-separated context, and they have reported a modest improvement over its static counterpart. \cite{Rebolini_2016,Rebolini_PhD} 
+Rebolini and Toulouse have performed a similar investigation in a range-separated context, and they have reported a modest improvement over its static counterpart. \cite{Rebolini_2016} 
 In these two latter studies, they also followed a (non-self-consistent) perturbative approach within the Tamm-Dancoff approximation with a renormalization of the first-order perturbative correction.
 %Finally, let us also mentioned the work of Romaniello and coworkers, \cite{Romaniello_2009b,Sangalli_2011} in which the authors genuinely accessed additional excitations by solving the non-linear, frequency-dependent eigenvalue problem. 
 %However, it is based on a rather simple model (the Hubbard dimer) which permits to analytically solve the dynamical equations.
@@ -757,11 +761,11 @@ In these two latter studies, they also followed a (non-self-consistent) perturba
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 \section{Conclusion}
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-Although far from being exhaustive, we hope to have provided, in the present \textit{Perspective}, a concise and fair assessment of the strengths and weaknesses of the Bethe-Salpeter equation (BSE) formalism of many-body perturbation theory.
+Although far from being exhaustive, we hope to have provided, in the present \textit{Perspective}, a concise and fair assessment of the strengths and weaknesses of the BSE formalism of many-body perturbation theory.
 To do so, we have briefly reviewed the theoretical aspects behind BSE, and its intimate link with the underlying $GW$ calculation that one must perform to compute quasiparticle energies and the dynamically-screened Coulomb potential; two of the key input ingredients associated with the BSE formalism.
 We have then provided a succinct historical overview with a particular focus on its condensed-matter roots, and the lessons that the community has learnt from several systematic benchmark studies on large molecular systems.
 Several success stories are then discussed (charge-transfer excited states and combination with reaction field methods), before debating some of the challenges faced by the BSE formalism (computational cost, triplet instabilities, lack of analytical nuclear gradients, ambiguity in the definition of the ground-state energy, and limitations due to the static approximation).
-We hope that, by providing a snapshot of the ability of BSE in 2020, the present \textit{Perspective} article will inspire the next generation of theoretical and computational chemists to roll up their sleeves and embrace this fascinating formalism, which, we believe, has a bright future within the physical chemistry community. \xavier{ Je serais un peu moins emphatique ... We hope that, by providing a snapshot of the ability of BSE in 2020, the present \textit{Perspective} article will motivate a larger community to   participate to the development of this alternative to TD-DFT which, we believe, may be extremely valuable within the physical chemistry community.}
+We hope that, by providing a snapshot of the ability of BSE in 2020, the present \textit{Perspective} article will motivate a larger community to participate to the development of this alternative to TD-DFT which, we believe, may become a very valuable computational tool for the physical chemistry community.
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